Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision Last revision Both sides next revision | ||
|
optimization_set [2016/05/20 22:04] nikolaj |
optimization_set [2016/07/24 21:21] nikolaj |
||
|---|---|---|---|
| Line 4: | Line 4: | ||
| | @#55CCEE: context | @#55CCEE: $ \langle Y, \le \rangle $ ... Non-strict partially ordered set | | | @#55CCEE: context | @#55CCEE: $ \langle Y, \le \rangle $ ... Non-strict partially ordered set | | ||
| | @#55CCEE: context | @#55CCEE: $ r:B\to Y $ | | | @#55CCEE: context | @#55CCEE: $ r:B\to Y $ | | ||
| - | | @#FF9944: definition | @#FF9944: $ O_r := \{\beta\in B\mid \forall(b\in X).\,r(\beta)\le{r(b)}\}$ | | + | | @#FF9944: definition | @#FF9944: $ O_r := \{\beta\in B\mid \forall(b\in B).\,r(\beta)\le{r(b)}\}$ | |
| ----- | ----- | ||
| - | If ${\mathrm{min}(r)}\subseteq B$ denote the minimum values of $r$, then | + | If ${\mathrm{min}(r)}\subseteq Y$ denote the minimum values of $r$, then |
| - | $O_r = R^{-1}({\mathrm{inf}(r)})$ | + | $O_r = r^{-1}({\mathrm{min}(r)})$ |
| with $r^{-1}:{\mathcal P}Y\to{\mathcal P}B$. | with $r^{-1}:{\mathcal P}Y\to{\mathcal P}B$. | ||
| Line 28: | Line 28: | ||
| (the indexed subspace of $X\to Y$ is called hypotheses space) | (the indexed subspace of $X\to Y$ is called hypotheses space) | ||
| - | and find from this set find the optimal fit (given by optimal $\beta\in B$) w.r.t. loss function $V:Y\to Y$ by optimizing | + | and find from this set find the optimal fit (given by optimal $\beta\in B$) w.r.t. loss function |
| + | |||
| + | $V:Y\times Y\to Y$ | ||
| + | |||
| + | by optimizing | ||
| $r(\beta):=V(f(\beta,x),y)$ | $r(\beta):=V(f(\beta,x),y)$ | ||
| Line 39: | Line 43: | ||
| with loss function | with loss function | ||
| - | $V(y',y)=(y'-y)^2$ | + | $V({\hat y},y)=({\hat y}-y)\cdot({\hat y}-y)$ |
| - | In practice, $x_i$ may be vectors and then $w$ is taken to be an inner product. | + | In practice, $x_i$ may be vectors and then $V$ is taken to be an inner product. |
| === Reference === | === Reference === | ||
